Semiotic analysis of the observer in relativity, quantum mechanics, and …€¦ · ·...

Vern S. Poythress

Semiotic analysis of the observer inrelativity, quantum mechanics, and apossible theory of everything

Abstract: Semiotic analysis of the role of the observer in the theory of relativityand in quantum mechanics shows the semiotic function of basic symmetries,such as symmetries under translation and rotation. How can semiotics berelevant to theories in physics? It is always human beings who form the theories.In the process of theory formation and communication, they rely on semioticsystems. Included among these systems is the semiotics involved in our pre-theoretical human understanding of space, time, and motion. Semiotic systemsthereby have an influence on theories in physics. As a result, key concepts infundamental physical theory have affinities with semiotics. In terms of KennethPike’s tagmemic theory, applied as a theory of theories, all symmetries take theform of distributional constraints. The additional symmetry under Lorentz trans-formations introduced by the special theory of relativity fits into the samepattern. In addition, constraints introduced by the addition of general relativitysuggest the form and limitations that might be taken by a “theory of everything”encompassing general relativity and quantum field theory.

Keywords: theory of relativity, quantum mechanics, symmetry, theory of every-thing, perspectives, tagmemics

DOI 10.1515/sem-2015-0006

A number of researchers have undertaken semiotic analysis of quantummechanics (Christiansen 1985, Christiansen 2003; Dosch et al. 2005a; Doschet al. 2005b; Januschke 2010; Prashant 2006). We wish to focus specifically onthe problem of the observer. Semiotic analysis is relevant to understanding therole of the observer, because observers are people. And people presuppose andinvoke semiotic systems as they form theories and communicate them.Moreover, in the twentieth century, physics was forced to wrestle with theinvolvement of observers in physical measurement, and this involvement

Vern S. Poythress, Westminster Theological Seminary – New Testament, 2960 W. Church Road,Glenside, PA 19038, USA, E-mail: [email protected]

Semiotica 2015; 205: 149–167

Bereitgestellt von | De Gruyter / TCSAngemeldet

Heruntergeladen am | 16.09.15 13:30

actually became a key element in the formation and explication of theories inphysics. Several distinct theories in physics now attribute a significant role tothe observer: the special theory of relativity, the general theory of relativity, andquantum mechanics (together with its successor, quantum field theory). As weshall see, key elements in the semiotics of space, time, and motion haveaffinities with what we see coming out in the structure of fundamental physics.These affinities are not an accident, because the scientists are tacitly reckoningwith semiotically structured knowledge in their theory formation.

1 Historical questions in quantum mechanics

The history of quantum mechanics shows the potential pertinence of semiotics.Nonrelativistic quantum mechanics received a stable mathematical formulationin the period 1925–1932 (Jammer 1966). Experimentally it was a success, but itentrained vexing questions about the nature of the observer’s involvement withreality, questions that continue to be debated (Krips 2013; Laudisa and Rovelli2013). Semiotics can analyze the nature of observer involvement. In the process,semiotically informed analysis may show how the scientific theories rely onsemiotic systems being used by the scientists.

2 Symmetry

Classical mechanics, special relativity, general relativity, and quantum mechanicsall utilize the concept of symmetry in their formulations. One of the most obvioussymmetries in fundamental theories of physics is the symmetry related to rotationsin space. A guiding assumption in theory formation is that fundamental physicallaws should look the same no matter which direction one looks. A physicist candescribe this property by saying that the laws are invariant under rotation.

Physical theories focus on the phenomena that they are investigating, not onthe observer as such. But we can see that invariance under rotation presupposesa role for the observer. The variable direction to be used as a result of a rotationis the direction that the observer chooses. Or we might call it the direction thatthe investigator chooses. We are here not reducing observers merely to readinginstruments and collecting data, as the concept of an “observer” sometimesoccurs in quantum mechanical discussions of an “observation” of a quantumsystem. We are considering the observers as thinkers and theorizers as well. Forthis reason, we undertake our analysis using a specific semiotic framework that

150 Vern S. Poythress



discusses observers, theories, and multiple theoretical perspectives (Pike 1976;Poythress 2013b). Our framework utilizes tagmemic theory as developed byKenneth L. Pike (1967, 1982a, 1982b; Pike and Pike 1977).

3 Perspectives

Tagmemic theory specifically invokes the use of three interlocking perspectivesor “views”: the particle view, the wave view, and the field view (Pike 1959; Pike1982; Poythress 2013b). The labels originated historically from Pike’s interactionwith physics and quantum mechanics (Pike 1959; Pike 1976: 99–100), but Pikeadapted the labels to his own meanings. They are not to be equated with the useof terms within quantum mechanics.

The particle perspective focuses on “particles” or items that can be contras-tively distinguished from other items around them. The fundamental probabil-istic relation characteristic of particle analysis is mutual exclusion. For particle-like semiotic units A and B, the probability P(A & B) of joint occurrence is near 0.If units A and B are mutually exclusive, P(A & B) ¼ 0. But in typical semioticcontexts involving human investigators, we must be content with approxima-tions. P(A & B) is near to 0, or approximately 0, which we write using theapproximation symbol “~”: P(A & B) ~ 0. The wave perspective focuses ongradual variation, either in time or in conceptualization. The fundamental prob-abilistic relation characteristic of wave analysis is inclusion: P(A | B) is near 1,when B includes or presupposes A or extends A. The third perspective, the fieldperspective, focuses on multidimensional relationships, often relationships orga-nized in cross-cutting arrays. The fundamental probabilistic relation character-istic of field analysis is probabilistic independence: P(A & B) ~ P(A) � P(B),which is equivalent to proportionality relations: P(A & B)/P(A) ~ P(not-A & B)/P(not-A), where not-A is the event where A does not occur.

The three perspectives were originally developed in the context of analysisof language, but can be applied more broadly in the context of semiotic analysis(Pike 1967; Waterhouse 1974; Poythress 1982a, Poythress 1982b, Poythress 2013a,Poythress 2013b).

4 The physics of “translations”

So let us consider the investigator in physics from a semiotic point of view. Fromthe particle perspective, the investigator can change his viewpoint by shifting toa new location. The new location is distinguishable from the old one: the two

Semiotic analysis of the observer 151



locations are mutually exclusive. This change is the kind of change that natu-rally gets classified as change of a particle kind. Linear measurements will takeplace with the new location treated as the origin. In terms of semiotic analysis,the investigator distinguishes the two locations and the two systems of measure-ments using the particle perspective.

In this context, the conception of changing location in the context of aphysical theory depends on the deeper, semiotic conceptions of the investiga-tor, who already has nontechnical intuitions about the nature of spatial loca-tion. Before the investigator becomes a theorist, he already has a pre-theoretical understanding of space and location. He has a heritage from hisculture and from the sense of space given through his senses. He can talkabout location through language or through gestures. Thus, space as concep-tualized by the investigator already has a semiotic structure. Moreover, the twolocations are distinguished emically, in a manner that is culturally meaningful,since the distinction is recognizable in an entire culture, not merely in a singleindividual.

Now consider a particular case of changing location. Suppose that measure-ments use three coordinate axes in three directions in space. The simplest formof change in location is a change in one coordinate axis, let us say the x-axis.Thus

xnew ¼ xold þ c

Physicists customarily write this change in a more abbreviated form, by letting asymbol such as x′ denote the new value and the symbol x denote the old value:

x 0 ¼ x þ c

From a semiotic viewpoint, the physicist has changed emic units, from an emicunit x in the old system to an emic unit x′ in the new system. If we regard thetwo systems as first-order “theories,” the change is a change of a particle kind inthe sense indicated in the earlier work on theories of theories (Poythress 2013b).

This change in the semiotic viewpoint in the investigator is the intuitivebasis for the theoretical concept of a translation in space. A translation in spaceis defined as a shift in the measurement system produced by placing the originfor measurement at a different location. The equation of transformation isprecisely x′ ¼ x þ c, if the translation is in the x-direction. General translationsmay shift in all three directions:

x 0 ¼ x þ cxy 0 ¼ y þ cyz 0 ¼ z þ cz




Physicists expect the fundamental laws with look the same after a translation.This property is called invariance under translation. It is of course a theoreticalconstraint in the minds of physicists. But before it takes on that status, it has tobe informally understood in the bodily experience of the investigator, and it hasto be grasped mentally by means of a tacit reliance on the particle perspective(on tacit knowledge, see Polanyi 1964, Polanyi 1967). Thus, the concept ofinvariance also has a semiotic basis.

In addition, the investigator is tacitly relying on the semiotics involved inwhat Kenneth Pike (1982; 1967: 86–87) termed a distributional constraint(Poythress 2013b: 95–96). The physical laws remain the same when the locationis changed by translation. Thus the location and the structure of the laws formtwo independent axes within a semiotic field.

5 Rotations

The investigator can also change his point of view not by moving to a newlocation, but by shifting his gaze, while remaining in the same location. Thisshift of gaze is the intuitive basis (pre-theoretical semiotic basis) for understand-ing the theoretical concept of rotation. Rotation automatically changes not onebut two or three of the values of the coordinates. The general form of a rotation is

x 0 ¼ a11x þ a12y þ a13z

y 0 ¼ a21x þ a22y þ a23z

z 0 ¼ a31x þ a32y þ a33z

where the values aij are constants. The constants have to be chosen in such away that lengths are preserved, and handedness is preserved (a figure shapedlike a right hand is rotated into a figure of the same shape and orientation).

From a semiotic viewpoint, a rotation produces a whole new emic set ofcoordinates, which interlock with the old ones in a complex way. This wholesalereplacement and wholesale interlock is a form of field-like change in the first-order“theory” that describes the physical system from only one fixed point of view. Thefundamental physical laws are supposed to remain the same under rotation.

6 Motion

A third kind of shift in perspective can occur if the investigator starts moving.The classic illustration, used by Albert Einstein (1920: 14), imagines two




observers. The first observer stands on the platform alongside a railroad track.The second observer is on a train moving along the track. The first observer, theone standing on the track, sees the train moving. From his point of view, thesecond observer is moving at the same speed as the train. And so is everythinginside the train. If the second observer, seated in the train, regards his ownposition as fixed, he sees the scenery and the first observer moving relative tohis position. That is, they are moving from his point of view.

So far, the situation with the train, and Einstein’s description of it, involverealities that ordinary human beings understand, apart from any scientifictraining. They can understand, and they can communicate. They communicatethrough language or diagrams or pictures or invitations for others to standalongside and experience the same observations. These common experiencesobviously depend on the semiotic structures involved in language, gestures,pictures, and the ideas of “commonality” in experience.

Einstein’s physical theory builds on these semiotically structured founda-tions. Einstein postulated that the two positions are equivalent from the stand-point of physical laws. If the observer on the train were to drop a ball, he wouldsee the ball fall straight down toward the floor, the same as it would if the trainwere not moving along the track at all. Thus, the observer cannot tell that thetrain is moving unless he looks outside the train. And if another train passes ona parallel track, he cannot tell which train is moving relative to the groundwithout looking at the ground or at distant scenery. Such is the principle ofrelativity: motion is relative to the observer.

This change in motion is a third kind of change in perspective. It involvescontinual small adjustments in the position of the origin for measurement, whenwe compare two investigators. The small changes are a form of wave-like changein perspective (Poythress 2013b). Einstein’s physical theory presupposes theunderlying reality of semiotically structured understanding of motion.

Suppose that the train is moving with a fixed velocity v in the x-direction.Then what is the difference between the way in which things will be measuredby the two observers, one on the platform and the other on the train? Beforethe work of Einstein, people could already conceive of such changes. Theywould have said that, as a result of the motion, the distance x′ measuredby the observer on the train in the direction of forward motion of the trainwould continually decrease relative to the distance measurement x by theobserver on the platform. In a short time period t, the train covers adistance of vt. At the end of the time, x′ is shortened by the amount vt relativeto x. Thus:

x 0 ¼ x � vt




Since the train is moving in the x-direction, the measurements in the directionsof the other coordinate axes do not change:

y 0 ¼ y;

z 0 ¼ z

7 Multiple theories

A physical theory such as Newtonian mechanics that describes interactions froma single observational framework is a kind of first-order theory. The observa-tional framework functions as a perspective on the physical world. But, as wehave seen, the conceptual capabilities and semiotic capabilities of human natureallow us to consider multiple perspectives, and we may theorize within asecond-order theory about the relationships between multiple perspectives.Traditional Newtonian mechanics in its more advanced forms already doesthis, and so it is a second-order theory from the semiotic point of view(Poythress 2013b: 89–96).

8 Galilean invariance

The investigator who is thinking about changes of perspective can rise above allthe particularities of various observers. Having used all three perspectives, hemay postulate that fundamental physical laws are invariant under translations(particle-like changes in perspective), rotations (field-like changes), and changesin velocity (wave-like changes). The third kind of change is also called a changein inertial system or inertial frame. The reference to inertia implies that the train(or other observation platform) must be moving at a constant velocity, ratherthan accelerating. An acceleration would be detectable, since the observer onthe train would feel it (more precisely, he would feel the train seat pushing onhim as it accelerates relative to the ground).

This threefold invariance – under translation, rotation, and motion at aconstant velocity – is sometimes called Galilean invariance, in honor of GalileoGalilei. The corresponding transformations between different observational fra-meworks are Galilean transformations (Goldstein 1980: 276).

The assumption of invariance is a powerful one; it constrains the form thatphysical laws must take. With only a few more assumptions, it allows us to inferNewton’s three laws of motion (Poythress 2013b: 89–96), which provide thebasic framework for classical mechanics. As in the case of invariance under




translation, the other kinds of invariance are distributional constraints(Poythress 2013b: 95–96).

9 The origins of the special theory of relativity

Albert Einstein worried about Galilean invariance, because it was not compati-ble with Maxwell’s equations for light and electromagnetic radiation. Galileaninvariance implied that the speed of light should change with the motion of theobserver (Goldstein 1980: 276–277), and that an observer going sufficiently fastshould be able to observe light in the form of standing waves rather than wavesin motion. Einstein (1920: 21–24) therefore undertook to see whether the princi-ple of Galilean invariance should be altered.

The process of alteration, we may observe, is a process of theory formation,and as such can be analyzed from a semiotic viewpoint. Einstein was motivatedby empirical concerns. But these concerns overlap with concerns about theobserver. Einstein (1920: 30–33) undertook a careful analysis of the process ofmeasurement, through which he showed that the ideal of absolute Newtoniansimultaneity in time, independent of all observers in all frames of reference,could not be observationally guaranteed. In particular, in order to synchronizeclocks, observers on the station platform and on the train had to send signals,and the signals could not propagate faster than the speed of light. This con-straint undermines any attempt to fix the measurement of time in a wayindependent of all observers.

An analogous conclusion might have been reached by observing from thestandpoint of semiotics that particle and wave perspectives are always inter-locked or entangled with one another. Investigators are always tacitly presup-posing a variety of perspectives, and the presuppositions become even moreevident when we erect a semiotic theory of theories, where we may be forced tobecome explicit about multiple perspectives in order to explain the multiplicityof possible theories about the same subject matter. If particle and wave perspec-tives are entangled, then in particular the particle perspective that differentiatesframes of reference through translation in space is entangled with the waveperspective that differentiates frames of reference through relative motion.Translated back into physics, that entanglement suggests that space and timemight be “entangled” rather than being perfectly separable in a manner inde-pendent of observer perspective.

The precise way that the two are entangled cannot be predicted withoutempirical observation. Since symmetries in translation and in rotation constrainthe shape of classical mechanics, one might search for some additional




symmetry that would constrain the shape of a revised form of mechanics, oncethe entanglement of space and time is allowed. Einstein postulated a particularsymmetry, namely the constancy of the speed of light in a vacuum. This con-stancy would be shared by all observers, and would be independent of the framein which an observer resided.

Given only this one additional assumption, Einstein was able to work outthe relationships between observers and between observations of time andlength made in two distinct inertial frames. This “working out” is the specialtheory of relativity.

Its key secret is a new “symmetry” – and symmetry is a characteristicdistributional aspect of semiotic theories of theories (Poythress 2013b: 95–96).Given this key new symmetry, Einstein’s new theory revises “Galilean invar-iance” to “Lorentz invariant” relativity, which again includes symmetries underrotation, under translation, and under changes in velocity. Lorentz invariancehas equations that match Galilean invariance at low velocities, but deviate moreand more as the velocity difference approaches the speed of light. This partialcoalescence with Galilean invariance represents an instance in theory-makingwhere a more complex theory (in this case, special relativity) includes a simpler,earlier theory (Newtonian mechanics) within it as a limit case (Poythress 2013b;also Dosch et al. 2005a: §5.1).

10 Quantum mechanics

Now consider elementary quantum mechanics. Symmetry principles play acentral role in quantum mechanics, as well as in classical Newtonian mechanicsand in the special theory of relativity. This use of symmetry again has an affinitywith the distributional properties of emic units in a semiotic system.

As an example, consider a single, isolated atom. Since an atomic nucleushas a mass much greater than the surrounding electrons, the nucleus can betreated as approximately fixed in space. A single atom is then rotationallysymmetric about its nucleus. The set of all possible rotational transformationswith the nucleus as a center forms a mathematical group, the three-dimensionalrotation group. The mathematical properties of the rotation group, together withthe group of symmetries under interchange of two or more electrons, constrainmany of the properties of the electron orbits and the atomic spectra related tothem (Wigner 1959).

We can be more specific about relationships between semiotics and quan-tum mechanics. A natural probabilistic framework for studying semiotic systems




treats as basic the probabilities for occurrence or non-occurrence of emic unitsA1, A2, A3,… at specific times and places. The probabilities are then probabilitiesof the form P(Ai), or more precisely P(Q(Ai)), where Q(Ai) is the question, “Doesthe eme Ai occur at the specified time and place?” P(Q(Ai)) is the probability,as estimated by the semiotic investigator, that Q(Ai) will receive a “yes” answer.

This kind of probability analysis seems superficially to be different intexture from the probabilities associated with quantum mechanics, since mea-surements in quantum mechanics typically concern continuous observables likeposition and momentum. But Mackey (1963: 64–71) has shown that quantummechanics can be reconfigured so that all questions about observables aretranslated into yes-no questions about whether the observable has a valuefalling within a (Borel) subset of the set of real numbers. Then all observablescan be derived (constructed) from questions with yes-no answers. Mackey’sreconfiguration is simply one form of an investigator’s decision to switch to anew suite of observables (Poythress 2013b: §3). This switch is in fact a conve-nient one for dealing with all observables in a uniform way, since some obser-vables (like the spin of elementary particles) cannot be treated as continuous.

Once the switch is made to questions, Mackey’s formal structure for quan-tum mechanics has a structure parallel to the structure of a probabilistic theoryof semiotics, with the questions in quantum mechanics being parallel to ques-tions about the occurrence of emic units in semiotics. In fact, since Mackey’squestions in the quantum mechanical context concern emic units within quan-tum theory, Mackey’s questions are in fact semiotic questions from the stand-point of the semiotician, as well as being quantum mechanical questions fromthe standpoint of Mackey’s view as a mathematical physicist. The usual symme-tries that we have discussed for Newtonian mechanics have analogues inMackey’s structure, and they correspond to distributional constraints withinsemiotic theory.

Mackey needs one key postulate to get a theory that represents quantummechanics rather than classical mechanics. The set of questions for quantummechanics, with suitable partial ordering, is isomorphic to “all closed subspacesof a separable, infinite dimensional Hilbert space” (Mackey 1963: 71). Mackeyunderstands that this additional postulate needs explanation, but the postulateis not as weird as it may seem at first. It is in fact the next simplest alternative tothe situation where the questions form a Boolean algebra, which would result inclassical mechanics. The decisive difference is that, in quantum mechanics, notall questions can be answered simultaneously. This restriction is an expressionof the Heisenberg uncertainty principle, which says that one cannot haveindefinitely precise measurements of both the position and the momentum ofa single particle.




One can find a suggestive analog to the basic principle of quantummechanics already lying within semiotics. As we saw above, the interlockingof perspectives within semiotics suggests the interlocking of the shifts in per-spective through translation and through change in velocity. The same principleof interlocking can be applied to the suite of observables that belong to a singleparticle. In classical mechanics the observables for a single particle includethree position observables (typically x, y, and z coordinates for the particle’slocation) and three velocity observables (typically vx, vy, vz, the velocity compo-nents in the x-, y-, and z-directions).

The interlocking of particle and wave perspectives suggests the interlockingof location measurements (location being particle-like in its contrasts with otherlocations) and velocity measurements (velocity being wave-like in that it intrin-sically involves comparison of neighboring values). This interlocking suggests inturn that there might be limits to simultaneous measurements of position andvelocity. These limits would be analogous to the limits that special relativitypostulates for observational simultaneity at high velocities (approaching thespeed of light).

The constraint on simultaneous measurement implies that the structure ofquestions cannot form a Boolean algebra. The most regular alternative to aBoolean algebra is what Mackey’s postulate chooses. Given this postulate, andthe usual symmetries assumed in classical mechanics, Mackey is able to develop,with few extra assumptions, the entire structure leading to Schrödinger’s equationand the results of nonrelativistic quantum mechanics. This result is in harmonywith semiotic analysis that stresses the importance of a semiotic form of symmetryas a distributional constraint (Poythress 2013b).

11 Combining relativity and quantum mechanics

When physicists reckon with the special theory of relativity, the symmetriescharacterizing the fundamental laws have to be adjusted. The relevant symme-tries include not only translations in space and time and rotations in space, butLorentz boosts (Einstein 1920; Resnick 1968: 60). The range of symmetries isexpressed in what is called the Poincaré group or the inhomogeneous Lorentzgroup, which plays a fundamental role in quantum field theory (Ryder 1996: 56).Judicious reasoning on the basis of these symmetries shows what are the viablepossibilities for spin for elementary particles, and how the behavior of spin ishighly constrained (Weinberg 1995: 49–106, 191–258). So even in this advancedform of quantum mechanics we can still see the influence of structures thatoriginate with the semiotics of space, time, and motion.




12 A theory of everything?

Physics now confronts the challenge of integrating fundamental theories into a“theory of everything.” Such a theory would include already established theoriesas first-order approximations. It would include the standard model for elemen-tary particle physics, which is an expression of quantum field theory, based onthe symmetries of special relativity and wave-particle duality in classical quan-tum mechanics. It would also include the general theory of relativity. Thisintegration has proved not to be easy. As of 2013, the leading candidate for atheory of everything seems to be string theory, but there are complaints (Smolin2007; Woit 2007), and some people are exploring alternatives.

The general theory of relativity postulates an additional invariant, beyondthe ones that we have discussed. Physical law should be invariant under achange from a gravitational field to an acceleration. In developing the generaltheory of relativity, Einstein pictured a box whose occupants could not seeoutside the box in which they were confined. He postulated that, if they felt aforce pulling them towards the floor, they would be unable to tell from anyinternal measurement whether the force was due to an acceleration in anelevator or to a gravitational field (Einstein 1920: 79–80). This constraint againdepends on the role of the observer–in this case, the observer inside the box.

Mathematically, this invariance between gravitation and acceleration can beexpressed by a transition to generalized coordinates, together with a generalizedrepresentation of the “metric” that measures distances using the chosen coordi-nate system. Generalized coordinates can be used not only for three spatialdimensions, but for the dimension of time as well, leading to a four-dimensionalmathematical representation (Einstein 1920: 116).

13 Generalized coordinates

Generalized coordinates have proved useful in physics in other contexts besidesgeneral relativity, so it is worthwhile reflecting on their semiotic significance.The choice of a coordinate system is up to the investigator, as we have seen indiscussing rotations and translations. Generalized coordinates function as oneversion of choice of a coordinate system, and thus fall under the semioticanalysis of theories of theories, and the semiotic analysis of multiple perspec-tives. Each choice of coordinate system is a choice of a suite of emic units,namely the units represented by the coordinate axes, together with a scale formeasurement (such as meters in space and seconds in time). The translations,




rotations, and Lorentz boosts involved in the special theory of relativity allinvolve linear equations of transformation, such as x′ ¼ x þ c for a translationof c units in the x-direction or

x 0 ¼ ax þ by

y 0 ¼ cx þ dy

for a rotation by a fixed angle about the z-axis. (For a rotation by a fixed angleθ, a ¼ cos θ, b ¼ sin θ, c ¼ − sin θ, and d ¼ cos θ.)

Generalized coordinates offer a distinct approach because the new coordi-nates need not be related to the old ones in a linear way. Nonlinearity meansthat the equations of motion in the new coordinate system may not look exactlylike the old equations. Indeed, according to the principle of general relativity,the measured values definitely will not look the same in an accelerated system,because acceleration is equivalent to gravitation.

Even prior to Einstein’s work on general relativity, generalized coordinateswere used in physics in situations where the use of a nonlinear transformationsimplified the mathematics or illumined the physical situation. The easiest case toillustrate is a case of cylindrical or spherical symmetry (note again the importanceof symmetry, as a semiotic property). Suppose an engineer is studying or model-ing the rotation of a cylindrical rod. The system is symmetrical around the centralaxis of the rod. So, instead of using the normal Cartesian coordinates x, y, and z, itmay prove convenient to use cylindrical coordinates. Starting with an arbitrarysystem of Cartesian coordinates, it is possible by translation to locate the origin ofthe system at one end of the central axis of the rod. Then by rotation one canmake the z-axis identical with the central axis of the rod. Finally, one introducesnew “cylindrical” coordinates, z, r, and θ. z, as before, measures the distance fromthe origin in the direction of the z-axis, the axis of the rod. r measures the radialdistance from any point to the z-axis, and θmeasures the angle between the x-axisand the radial direction leading to the point in question. (See Figure 1.)

The coordinates r and θ are related to the older coordinates x and y in anonlinear way, and the equations of motion in the new system of coordinateswill look different. But the equations within the new system of coordinates mayalso be revealing, possibly by having no dependence on θ. Because of thecylindrical symmetry, the angle θ should not affect the behavior of the rod orof particles interacting with the rod. Likewise, when a system such as a ball or asingle atom exhibits spherical symmetry, it is customary to using sphericalcoordinates r, θ, and φ, where r is the radial distance from the central point ofsymmetry and θ and φ measure angles between the radius and suitable fixedrays or planes.




Within classical mechanics, when we use these nonlinear coordinate systems,the standard elementary Newtonian equation F ¼ ma does retain its standardform. But physicists have found that for many physical systems it was possibleto write the equations in an invariant form. In fact, there are two differentinvariant forms, the Lagrangian and the Hamiltonian, depending on whetherwe choose to use generalized velocities or generalized momenta in addition togeneralized position coordinates (Simon 1960: 365–368, 396–399). These formu-lations show the role of symmetries in classical mechanics, and by doing soaffirm in addition the key role of distributional constraints in semiotic analysesof the theories.

However, in both the Lagrangian and Hamiltonian formulations within classi-cal mechanics, time normally plays a very distinct, special role. Einstein’s generaltheory of relativity removes this constraint: the fundamental equations treat time in

Figure 1: Cylindrical coordinates.




a manner parallel to space, and indeed they must do so, in order to do justice to thefact that the concept of simultaneity and the measurement of the rate of passing oftime vary with observer viewpoint.

14 Expectations for integration

The general theory of relativity includes the special theory of relativity as lesscomprehensive, special theory for dealing with “inertial frameworks” for mea-surement. The special theory serves as a first-order approximation for the gen-eral theory.

By analogy, a “theory of everything” would encompass both the generaltheory of relativity and the standard model for relativistic quantum mechanicswithin a more comprehensive theory, which would have these subordinatetheories as first-order approximations. But it would not literally be a theory ofabsolutely everything because it needs an investigator or theorist. The theoristcan rise above both the more comprehensive theory and the simpler theoriesencompassed by it, and can articulate the semiotic relations between the the-ories (Poythress 2013b). Gödel’s results concerning incompleteness in arithmetic(Nagel and Newman 2008) suggest by analogy that no theory sufficiently com-plex to include arithmetic can provide robust resources for theorizing about itselfwithout engendering contradiction or paradox. The same is suggested by thesemiotic analysis of theories, according to which we can postulate an indefi-nitely ascending series of theories, each of which is analyzing the theories belowit in the series.

In a semiotically oriented theory of theories, the symmetries and invariantsof physical theories are analyzed as a distributional constraint. But distributionis entangled with contrast and variation. The independence of observablespostulated as a distributional ideal is indeed an ideal, only approximated andnot actually completely realized in real systems. This entanglement suggests alesson for physical theory, namely that the invariances postulated by funda-mental theories will be approximate rather than exact. Elementary particletheory is already familiar with this approximate invariance in the form of“symmetry breaking.” Special relativity turns out to be an approximation,since no real physical system is ever completely isolated from distant gravita-tional interaction.

The search for a theory of everything theorizes that general relativity in turnis only an approximation. Might it be the case that the supposed equivalencebetween acceleration and gravity is only approximate? Or we could consider




even more radical deviations from symmetry. Might it be that the invariance ofphysical theory under translation or rotation is only approximate?

That sounds like an outlandish proposal, until one realizes that no smallphysical system is completely isolated from the rest of the matter/energy in theuniverse. Cosmologists postulate a roughly uniform spread of matter/energy forthe sake of simplicity in their models. But the distribution of matter in theuniverse is not completely uniform either under translation or rotation. So atthe level of a theory encompassing the entire universe, no real invariance exists.One must postulate an invariance for the fundamental equations, but not for thedistribution of matter/energy that the equations are designed to describe. Sincethe equations can only be tested with reference to the actual universe, which isnot completely symmetrical, there is no final way of deciding whether failure insymmetry in test results is due to the asymmetry of the universe or to a failure ofthe equations to include asymmetry in the laws. Thus, a theory of “everything”fails literally to be absolutely final because of entanglement between theory andconstraints in observation.

Or, to put in a way that is oriented to the investigator, the investigator cannever honestly eliminate alternative hypotheses, because he cannot eliminatemultiple perspectives on the meaning of his investigation. This limitations has asemiotic dimension: semiotics includes the potential for multiple perspectives.

15 Discrete or continuous

In trying to develop a “theory of everything,” researchers confront the questionwhether the ultimate “stuff” of the universe is discrete or continuous. Thecurrently favored option of string theory postulates that the ultimate nature ofthings should be represented by the spatial structure of a multidimensionalmanifold, which mathematically is continuous, not discrete. However, minorityoptions include discrete models, where, for instance, the ultimate constituentsare discrete quantum computations (Lloyd 2006; Lloyd 2007) or discrete causalstructures (Markopoulou 2000a, Markopoulou 2000b). Let us consider the ques-tion from the standpoint of tagmemic theory, treated as a semiotic theory.

In tagmemic theory, the particle view treats semiotic systems as discretecollections of particles, while the wave view treats the same systems as contin-uous waves that develop and interact. These two viewpoints interlock; more-over, each is equally ultimate (Pike 1982: 19–29).

Now consider current physical theory. Almost all current theories in physicsuse differential equations, which presuppose a backdrop of continuous spaceand time. This is true even in solid state physics, where the researchers know




full well that the solid state is discrete at the atomic level. But they customarilyrepresent it at the macrolevel using a continuous model, for the sake of simpli-city and solvability.

Mathematically, the representation of a continuum in space or time pre-supposes points on the continuum, and from an observer point of view eachpoint is discrete. Conversely, each discrete point is identifiable in terms of itslocation within a continuum. So, observationally speaking, we can see a mutualdependence. Newtonian mechanics embodies a physical form of this mutuality.Discrete point particles move in continuous space, and are identifiable partly bylocation. As in other cases, the understanding of both discreteness and conti-nuity relies on pretheoretical experience, which involves semiotic structureapplied to space and time.

The rise of quantum mechanics partially dissolves the discreteness of parti-cles, because subatomic “particles” are not completely localized in space.Quantum field theory further dissolves the materiality of “particles” becausethey pop in and out of existence, and the times for their existence are notcompletely localized. Yet the theory still retains indispensable ties with observa-tion, and observation still has the form of discrete events located within a space-time framework that is modeled as continuous. The theory includes discreteobservables like spin, and discrete energy states in the atom, but the mathema-tical models still have their basis in the mathematics of the continuum.

The interlocking of discrete and continuous within the world of the observersuggests that any theory that ultimately gets tested by means of correlationsbetween theory and observation must contain within itself a mutual dependenceof the discrete and the continuous. One can ask whether this mutuality ismodeled to some extent within quantum mechanics by the ability to shift froman analysis based on waves, using momentum eigenstates, and an analysisbased on particle position, using position as the observable distinguishing thestates. A theory of everything necessarily encompasses ordinary quantummechanics, as a first-order approximation. So the interlocking of discrete andcontinuous, and the ability to express the theory in terms of both kinds ofobserver viewpoint, would be desirable.

16 Conclusion

We cannot dictate what form a final theory would take, because as observers wedeal with a world outside ourselves. At the same time, we as observers areinformed by semiotic structures. The forms of interlocking in these structures




suggest useful heuristic constraints on the final theory. In addition, semioticanalysis makes visible ways in which semiotic structures inform existing the-ories in physics.

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Bionote

Vern S. Poythress

Vern S. Poythress (b. 1946) is a professor at Westminster Theological Seminary. His researchinterests include hermeneutics, mathematical linguistics, and theology. His publicationsinclude The gender-neutral Bible controversy (2000); Redeeming science (2006); In thebeginning was the word: Language – a God-centered approach (2009); and Redeemingsociology (2011).




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